Tidbit 1
The writing workshops have made me refreshingly more attuned to things like this Tom's of Maine toothpaste tube:
In case you can't read the blurry photo, the text says (in part):
"Embrace the Power of Mother Nature! At Tom's of Maine that means we use authentic and effective natural ingredients.[1] We use zinc citrate sourced from zinc--a naturally occurring mineral and xylitol, a natural ingredient derived from birch bark or corn.[2]"
[1] Did you catch the missing comma after the introductory element? Tolkien left those commas out all the time, which I'm told is common in British English. Since Tom's of Maine is located in Maine, U.S.A., I hold the company to American English standards for comma usage. Aside from that, can anyone tell me what an "inauthentic" ingredient would be?
[2] I think they meant "We use zinc citrate (synthesized from naturally-occurring zinc ore) and xylitol (a natural ingredient derived from birch bark or corn)," but hey, I'm not a chemist. By the way, isn't saying "we use zinc citrate sourced from naturally-occurring zinc" kind of like saying "we use water sourced from naturally-occurring hydrogen and oxygen"?
Tidbit 2
While driving home the other day, I happened to observe that if the sum of the digits in a number is a multiple of 9, the number itself is evenly divisible by 9 (an extension of the similar tidbit about numbers being divisible by 3 if the sum of their digits is divisible by 3). This is old news to mathematicians, but brand new news to me. I don't know about you, but I feel empowered knowing, without doing any long division, that one trillion four billion seven hundred thousand five hundred one is divisible by 9.
My mathematician dad would want me to understand why this works, as opposed to simply knowing it works. I never appreciated that attitude when I was a kid bumming help from him on my math homework--in fact, I distinctly resented it--but I appreciate it now. So here you go, dad:
One trillion four billion seven hundred thousand five hundred one
= 1,004,700,501
= 1*(1+999,999,999) + 4*(1+999,999) + 7*(1+99,999) + 5*(1+99) + 1
= 1 + 4 + 7 + 5 + 1 + (a bunch of numbers divisible by 9).
Thus, if 1 + 4 + 7 + 5 + 1 is divisible by 9 (it is) then 1,004,700,501 is too.
In case you're wondering, I was thinking about nines because I've been doing a lot of Ken-Ken recently, a habit that started as a way to procrastinate finishing the Lord of the Rings trilogy (which, incidentally, I finished on Dec. 31, 2010).
5 comments:
So where is the general proof? One example, even if a bit complicated, does not a proof make. In your example, you do show the general idea behind the proof. Still a tough old guy,
Aging Reader
Sorry, I don't know how to type sigmas and all that into the blog, so instead I'll explain it using words. If you agree that any number "...dcba" can be represented as the sum of a ones, b tens, c hundreds, d thousands, etc.; and that 1 = 1, 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1, etc.; and that n(q+1) = nq + n; and that the sum of a number evenly divisible by x plus another number evenly divisible by x is also evenly divisible by x; and if you agree that if q is evenly divisible by x then nq is evenly divisible by x (assuming we're talking integers only); lessee, did I miss anything?; well, then you say a number ...vutsrqponmlkjihgfedcba can be represented as the sum of the individual digits times 10^z where z = the distance of the digit's position from the right end of the number, and that 10^z - 1 is a bunch of 9s in a row (z - 1 9s in a row, except if z = 0, in which case you get 1 - 1 = 0 which you can disregard because p + 0 = p), and that a number made of z - 1 9s in a row is divisible by 9 (i.e. 9*[z - 1 1s in a row]; then you combine all of those things you agreed to above (assuming you agreed to them), and you end up with the sum of a bunch of numbers divisible by 9 (all those z - 1 9s in a row numbers) plus the sum of all the individual digits that were in the original number in question because you multiplied them by the 1s you subtracted from the 10^zs. Does that work for you?
By the way, don't take that explanation as a gold standard for math writing. It's probably sacrilegious for the daughter of a mathematician to confess it never occurred to her--at least until she was well beyond the point when it would have been useful had it occurred to her--that people actually read the words in math textbooks. I thought the words were just an editorial binding agent to glue formulas together into textbook chapters. Sorry.
That's the idea. I guess I'd use more symbols but you have the idea. What does Elias think of your explanation?
Aging Reader
Changing the subject back to the non-tidbit part,
I agree that Chichester Psalms is fabulous. I got to accompany the last movement of it once, only on piano but it was still a joy to do. Wish I could be there ...
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